Puzzle for September 22, 2022 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
In eq.5, replace C with D + F (from eq.3a): D + E = A + D + F + F which becomes D + E = A + D + 2×F Subtract D from each side of the equation above: D + E - D = A + D + 2×F - D which becomes eq.5a) E = A + 2×F
Hint #2
In eq.6, replace E with A + 2×F (from eq.5a): A + D = C - D + A + 2×F In the above equation, subtract A from both sides, and add D to both sides: A + D - A + D = C - D + A + 2×F - A + D which becomes eq.6a) 2×D = C + 2×F
Hint #3
In eq.6a, substitute D + F for C (from eq.3): 2×D = D + F + 2×F which becomes 2×D = D + 3×F Subtract D from each side of the above equation: 2×D - D = D + 3×F - D which makes D = 3×F
Hint #4
Substitute 3×F for D in eq.3: C = 3×F + F which makes C = 4×F
Hint #5
Substitute 3×F for D, and 4×F for C in eq.4: E - 3×F = 4×F - E Add 3×F and E to both sides of the equation above: E - 3×F + 3×F + E = 4×F - E + 3×F + E which simplifies to 2×E = 7×F Divide both sides by 2: 2×E ÷ 2 = 7×F ÷ 2 which makes E = 3½×F
Hint #6
Substitute 3×F for D, 3½×F for E, and 4×F for C in eq.5: 3×F + 3½×F = A + 4×F + F which becomes 6½×F = A + 5×F Subtract 5×F from each side of the equation above: 6½×F - 5×F = A + 5×F - 5×F which makes 1½×F = A
Hint #7
Substitute 3×F for D, and 1½×F for A in eq.2: 3×F = 1½×F + B Subtract 1½×F from each side of the above equation: 3×F - 1½×F = 1½×F + B - 1½×F which makes 1½×F = B
Solution
Substitute 1½×F for A and B, 4×F for C, 3×F for D, and 3½×F for E in eq.1: 1½×F + 1½×F + 4×F + 3×F + 3½×F + F = 29 which simplifies to 14½×F = 29 Divide both sides of the above equation by 14½: 14½×F ÷ 14½ = 29 ÷ 14½ which means F = 2 making A = B = 1½×F = 1½ × 2 = 3 C = 4×F = 4 × 2 = 8 D = 3×F = 3 × 2 = 6 E = 3½×F = 3½ × 2 = 7 and ABCDEF = 338672